Theoretical Continuous Equation Derived from the Microscopic Dynamics for Growing Interfaces in Quenched Media

TL;DR

This paper derives an analytical continuous equation from microscopic rules for a model of growing interfaces in quenched media, revealing a form similar to QKPZ with multiplicative noise and matching known scaling exponents.

Contribution

The authors derive a new continuous equation from microscopic dynamics that naturally includes the nonlinear term and differs from the standard QKPZ equation.

Findings

Equation reproduces the scaling exponents of the model

Nonlinear term arises naturally from microscopic rules

Equation resembles QKPZ with multiplicative quenched and thermal noise

Abstract

We present an analytical continuous equation for the Tang and Leschhorn model [Phys. Rev A {\bf 45}, R8309 (1992)] derived from his microscopic rules using a regularization procedure. As well in this approach the nonlinear term $(\nabla h)^2$ arises naturally from the microscopic dynamics even if the continuous equation is not the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. {\bf 56}, 889 (1986)] with quenched noise (QKPZ). Our equation looks like a QKPZ but with multiplicative quenched and thermal noise. The numerical integration of our equation reproduce the scaling exponents of the roughness of this directed percolation depinning model.